In the third regime, there are several examples from the literature (eg. [1]}, [2]}, [3]}, [4]}), but they are all real analytic solutions. Without the real analytic assumption, due to the spatially non-consistent parabolic behavior, the existence is usually false and the uniqueness is unknown.
To address this gap, this paper studies to what extent the solution of \((\ref {muskat equation})\)  is analytic.
